All of the aforementioned researches have considered availability of perfect CSI, while due to the passive nature of the IRS, channel acquisition in IRS-aided systems is quite challenging, especially for the IRS-related channels. Although, some papers have presented robust beamforming designs for IRS-aided MIMO systems [1]}, [2]}, [3]}, [4]}, [5]}, to the best of our knowledge, no one has ever considered the effect of channel uncertainty in IRS-assisted NOMA and IRS-aided OMA systems.
To address this issue, in this paper a robust design for IRS-assisted NOMA communication system is devised. Motivated by [1]}, the sum-rate maximization problem is reformulated into a tractable optimization problem using the penalty dual decomposition (PDD) scheme [7]} and by applying the block successive upper bound minimization/maximization (BSUM) method [8]}, the problem is solved through an iterative algorithm. The solution in each iteration would be closed-form which results in very low computational complexity. black We consider two cases of continuous and discrete phase shifters in the IRS. In the literature, the authors either used exhaustive search to find optimal discrete phase shifts for the IRS, which requires huge computational complexity, or simply quantized the obtained solution for a continuous IRS, which increases performance loss. In this paper, we propose a smart phase selection for a discrete IRS, using a trellis-based algorithm.  The performance of the robust IRS-aided NOMA is compared with the non-robust and the perfect CSI scenarios in both cases of continuous and discrete IRS phase shifts.
blackWe, then, establish a comparison benchmark by providing a solution for maximizing the sum-rate maximization in an IRS-aided OMA system.To this end, we present a robust IRS-aided OMA design using FDMA and TDMA. By comparing NOMA and OMA in three cases of perfect CSI, imperfect CSI robust design and imperfect CSI non-robust design, we provide a new understanding of IRS-aided OMA and IRS-aided NOMA systems and their performance in different situations. 
To better clarify, the contributions of this paper are summarized as follows:
